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Clemson University

1. Clevenger, Thomas Conrad. A Parallel Geometric Multigrid Method for Adaptive Finite Elements.

Degree: PhD, Mathematical Sciences, 2019, Clemson University

Applications in a variety of scientific disciplines use systems of Partial Differential Equations (PDEs) to model physical phenomena. Numerical solutions to these models are often found using the Finite Element Method (FEM), where the problem is discretized and the solution of a large linear system is required, containing millions or even billions of unknowns. Often times, the domain of these solves will contain localized features that require very high resolution of the underlying finite element mesh to accurately solve, while a mesh with uniform resolution would require far too much computational time and memory overhead to be feasible on a modern machine. Therefore, techniques like adaptive mesh refinement, where one increases the resolution of the mesh only where it is necessary, must be used. Even with adaptive mesh refinement, these systems can still be on the order of much more than a million unknowns (large mantle convection applications like the ones in [90] show simulations on over 600 billion unknowns), and attempting to solve on a single processing unit is infeasible due to limited computational time and memory required. For this reason, any application code aimed at solving large problems must be built using a parallel framework, allowing the concurrent use of multiple processing units to solve a single problem, and the code must exhibit efficient scaling to large amounts of processing units. Multigrid methods are currently the only known optimal solvers for linear systems arising from discretizations of elliptic boundary valued problems. These methods can be represented as an iterative scheme with contraction number less than one, independent of the resolution of the discretization [24, 54, 25, 103], with optimal complexity in the number of unknowns in the system [29]. Geometric multigrid (GMG) methods, where the hierarchy of spaces are defined by linear systems of finite element discretizations on meshes of decreasing resolution, have been shown to be robust for many different problem formulations, giving mesh independent convergence for highly adaptive meshes [26, 61, 83, 18], but these methods require specific implementations for each type of equation, boundary condition, mesh, etc., required by the specific application. The implementation in a massively parallel environment is not obvious, and research into this topic is far from exhaustive. We present an implementation of a massively parallel, adaptive geometric multigrid (GMG) method used in the open-source finite element library deal.II [5], and perform extensive tests showing scaling of the v-cycle application on systems with up to 137 billion unknowns run on up to 65,536 processors, and demonstrating low communication overhead of the algorithms proposed. We then show the flexibility of the GMG by applying the method to four different PDE systems: the Poisson equation, linear elasticity, advection-diffusion, and the Stokes equations. For the Stokes equations, we implement a fully matrix-free, adaptive, GMG-based solver… Advisors/Committee Members: Timo Heister, Qingshan Chen, Leo Rebholz, Fei Xue.

Subjects/Keywords: Geometric multigrid; Parallel computing; Stokes equations

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APA (6th Edition):

Clevenger, T. C. (2019). A Parallel Geometric Multigrid Method for Adaptive Finite Elements. (Doctoral Dissertation). Clemson University. Retrieved from https://tigerprints.clemson.edu/all_dissertations/2523

Chicago Manual of Style (16th Edition):

Clevenger, Thomas Conrad. “A Parallel Geometric Multigrid Method for Adaptive Finite Elements.” 2019. Doctoral Dissertation, Clemson University. Accessed January 25, 2020. https://tigerprints.clemson.edu/all_dissertations/2523.

MLA Handbook (7th Edition):

Clevenger, Thomas Conrad. “A Parallel Geometric Multigrid Method for Adaptive Finite Elements.” 2019. Web. 25 Jan 2020.

Vancouver:

Clevenger TC. A Parallel Geometric Multigrid Method for Adaptive Finite Elements. [Internet] [Doctoral dissertation]. Clemson University; 2019. [cited 2020 Jan 25]. Available from: https://tigerprints.clemson.edu/all_dissertations/2523.

Council of Science Editors:

Clevenger TC. A Parallel Geometric Multigrid Method for Adaptive Finite Elements. [Doctoral Dissertation]. Clemson University; 2019. Available from: https://tigerprints.clemson.edu/all_dissertations/2523

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